A Givens rotation based matrix decomposition method has been proposed as a way of building a Multiple-Input-Multiple-Output (MIMO) preceding matrix for IEEE 802.16e Mobile WirelessMAN standard (see IEEE C802.16e-04/516, Nortel Networks: Unified MIMO Pre-Coding based on Givens Rotation, 4 Nov. 2004 or J. C. Roh and B. D. Rao, “An Efficient Feed-back Method for MIMO Systems with Slowly Time-Varying Channels”, Proc. of IEEE Wireless Communications and Networking Conference (WCNC) 2004, Atlanta, Ga., March 2004).
For a singular value decomposition (SVD) based MIMO preceding technique, the mobile station (MS) is required to send a beam-forming matrix V to the base station (BS). Based on the unitary structure of the matrix V, the number of required parameters to represent V can be greatly reduced by utilizing Givens decomposition method.
For example, VεCt×n consists of tn complex numbers, which means it has 2tn real numbers as its elements. By using Givens decomposition, V can be represented by (2t−1)−n2 real numbers.
The Givens parameters can be further quantized by using a 1 bit scalar adaptive delta modulation (ADM) to allow further reduction of the redundancy in a time and/or frequency domain.
The ADM encoder quantizes the difference between a newly incoming sample and a previously quantized sample into 1 bit information. The ADM approach is designed to trace a slowly varying signal, but it is vulnerable to abrupt changes of the signal.
A channel decomposition method and the corresponding Givens parameter extraction method can be found in Roh and Rao's article cited above.
The ADM is an efficient scheme to track the correlated signal utilizing a limited resource. Tracking the Givens parameters, which are angular values, apparently is an appropriate method since they have bounded values, e.g., phase φε(−π, π], and rotational angle θε[0, π/2) (see Roh and Rao's article cited above), and as there exists correlation over time or frequency (in case of a multi-carrier transmission scheme like OFDM).
However, observation of progress of the phase value φ reveals the fact that there are discontinuities when the phase value φ approaches a border (π or −π). The phase value φ disappears at a certain point and promptly re-appears on the other side of the border. This behavior stems from the fact that π and −π are equivalent in terms of an angular value.
However, such effect bears the problem that even with a slowly varying signal, the bounded value representing it into a scalar can show abrupt changes, which may fail the effort of the ADM encoder to appropriately encode the signal differences.
This cyclic overflow, which is caused by the modular feature of the phase valueθ=θ+2πn, where θε(−π, +π] and n is integer,should be taken into account when designing an ADM encoder and/or decoder in case of tracing the phase value φ of the Givens parameters.
FIG. 1 and FIG. 2 show the typical behavior of the phase value φ and the rotational angle θ over time, for continuous phase values φ and for discontinuous phase values φ, respectively.
Input phase values φ 101, 201 and rotational angles θ 102, 202 are acquired by extracting Givens parameters in decomposing a right unitary matrix of a 2×1 MISO channel.
Urban Macro channel realizations, which are generated by an extended spatial channel model (SCMe), are used for simulation purposes. The MS speed is exemplary set to 10 m/s.
In case of tracking the rotational angle θ, which varies continuously within a range [0, π/2), the known ADM scheme is good enough to trace the input signal. Such is the case for continuously varying phase value φ according to FIG. 1.
In case of tracking the phase value φ in the presence of discontinuities 203, 204 according to FIG. 2, the known scheme loses track of the signal whenever a discontinuity 203 or 204 occurs, and it leads to fluctuations after this point. The tracking concept needs a considerable time to re-trace the input signal. Hence, system performance deteriorates due to the cyclic overflow event.
The adaptive delta modulation (ADM) is widely used to quantize a slowly varying scalar value. In slowly time-varying channels, the corresponding Givens parameters are also slowly and mostly continuously changing, which makes the ADM eligible to quantize Givens parameters.
An ADM encoder includes an accumulator and a one-bit quantizer. The working principle of ADM is described in M. A. Aldajani and A. H. Sayed, “A Stable Structure for Delta Modulation with Improved Performance”, Proc. of ICASSP, Salt Lake City, Utah, May. 2001.
A signal {circumflex over (φ)}[k] may be constructed that tracks a signal φ[k]. This can be achieved according to the following approach: At each instant of time, a start value {circumflex over (φ)}[k−1] is updated to {circumflex over (φ)}[k] so that this new value is closer to φ[k] than its previous value. Each update is based on the difference (or error) between φ[k] and {circumflex over (φ)}[k−1], defined byeα[k]=φ[k]−{circumflex over (φ)}[k−1].  (1)
The signal {circumflex over (φ)}[k−1] is increased or decreased by a positive amount Δ[k] depending on an encoder's output history and on a sign of the error (1).
A step-size Δ[k] of the one-bit quantizer is adaptively changing in order to better track the dynamics of the signal. The step-size is increased if two subsequently encoded bits are the same, and it is decreased otherwise, that is,
                              Δ          ⁡                      [            k            ]                          =                  {                                                                      α                  ⁢                                                                          ⁢                                      Δ                    ⁡                                          [                                              k                        -                        1                                            ]                                                                                                                                        if                    ⁢                                                                                  ⁢                                          c                      ⁡                                              [                        k                        ]                                                                              =                                      c                    ⁡                                          [                                              k                        -                        1                                            ]                                                                                                                                                                1                    α                                    ⁢                                      Δ                    ⁡                                          [                                              k                        -                        1                                            ]                                                                                                                                        if                    ⁢                                                                                  ⁢                                          c                      ⁡                                              [                        k                        ]                                                                              ≠                                      c                    ⁡                                          [                                              k                        -                        1                                            ]                                                                                                                              (        2        )            with
Δ[k] being the step-size;
c[k]=sign[eα[k]]ε{−1, +1} being an encoded bit for the k-th sample;
α being a system parameter, which satisfies α>1.
The sign of the error eα[k] according to (1), decides whether {circumflex over (φ)}[k−1] increases or decreases at each time instant.
Thus, the signal {circumflex over (φ)}[k] is varied according to an adaptation rule:{circumflex over (φ)}[k]={circumflex over (φ)}[k−1]+sign[eα[k]]Δ[k].  (3)
Observing the step-size Δ[k] reveals the following equivalent form.Δ[k]=αw[k]Δ[0]  (4)withw[k]=w[k−1]+q[k]  (5)andq[k]=c[k]c[k−1].  (6)
This alternative representation allows describing the scheme for updating {circumflex over (φ)}[k] in a block diagram according to FIG. 3 thereby depicting a known ADM encoder. The upper and lower parts of the figure implement equations (4) and (3), respectively.
It is to be noted that system parameters α and Δ[k] are known to the ADM encoder as well as to an ADM decoder, and only the encoded bit c[k] is required for the receiver to decode the quantized value {circumflex over (φ)}[k].
The ADM is a low-rate scalar quantization scheme (as low as one bit per parameter), and it has inherently a channel tracking feature for slowly varying channels.
The disadvantages of this method, however, are as follows: The ADM is an appropriate scheme to trace a slowly varying signal, but in case of abrupt transitions as expected with the phase value φ, it is likely to lose track of an incoming signal, in particular to fluctuate. In addition, the known ADM approach requires a considerable amount of time to re-trace the signal (see FIG. 2).
A difference between the actual phase φ and its quantized value {circumflex over (φ)} may be determined in terms of an average angular distortion (AAD):
                                          e            _                    ϕ                =                              𝔼            ⁡                          [                              e                ϕ                            ]                                =                                    1                              N                s                                      ⁢                                          ∑                i                                  N                  s                                            ⁢                              min                ⁡                                  (                                                                                                                                                                ϕ                            ^                                                    i                                                -                                                  ϕ                          i                                                                                                            ,                                                                                                                  2                          ⁢                                                                                                          ⁢                          π                                                -                                                                                                                                                                      ϕ                                ^                                                            i                                                        -                                                          ϕ                              i                                                                                                                                                                                                            )                                                                                        (        7        )            where Ns is the number of samples taken for the average value calculation.
The AAD value ēφ of a known ADM encoder with α=1.5 and Δ[0]=0.05 taken for Ns=60,000 samples amounts to 0.2409, in case of an Urban Macro channel with an assumed speed of the mobile station (MS) amounting to 10 m/s.
The problem to be solved is to overcome the disadvantages as stated before and in particular to provide an efficient approach as how to handle cyclic overflow events.
To overcome this problem, a method is provided for data processing, in particular for signal processing and/or for signal modulation, that includes:                tracking a phase parameter;        determining a discontinuity; and        compensating for the discontinuity.        
This approach advantageously allows in particular an adaptive delta modulation (ADM) encoder to handle a cyclic overflow event when tracking the phase value φ that is in particular one of the Givens parameters that is advantageously utilized for tracing a MIMO channel over time and/or frequency.